Reducing Matrix Polynomials to Simpler Forms
Zaballa, Ion
Jul 8, 2017

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Banff International Research Station for Mathematical Innovation and Discovery

Date Issued

2017-07-08T10:00

Description

A square matrix can be reduced to simpler form via similarity transformations.
Here ``simpler form'' may refer to diagonal (when possible), triangular (Schur),
or Hessenberg form. Similar reductions exist for matrix pencils if we consider
general equivalence transformations instead of similarity transformations.
For both matrices and matrix pencils, well-established algorithms are available
for each reduction, which are useful in various applications.
For matrix polynomials, unimodular transformations
can be used to achieve the reduced forms but we do not have a practical way to compute them.
In this work we introduce a practical means to reduce a matrix polynomial with
nonsingular leading coefficient to a simpler (diagonal, triangular, Hessenberg)
form while preserving the degree and the eigenstructure.
The key to our approach is to work with structure preserving
similarity transformations applied to a linearization of the matrix polynomial
instead of unimodular transformations applied directly to the matrix
polynomial. As an applications, we will illustrate how to use these reduced forms
to solve parameterized linear systems.

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